Page 85 - Materials Chemistry, Second Edition

72 2 Solid-State Chemistry

Figure 2.43. Comparison between a real monoclinic crystal lattice (a ¼ b 6¼ c) and the corresponding
reciprocal lattice. Dashed lines indicate the unit cell of each lattice. The magnitudes of the reciprocal
lattice vectors are not in scale; for example, |a*| ¼ 1/d 100 , |c*| ¼ 1/d 001 ,|G 101 | ¼ d 101 , etc. Note that for
orthogonal unit cells (cubic, tetragonal, orthorhombic), the reciprocal lattice vectors will be aligned
parallel to the real lattice vectors. # 2009 From Biomolecular Crystallography: Principles, Practice,
and Application to Structural Biology by Bernard Rupp. Reproduced by permission of Garland Science/
Taylor & Francis Group LLC.

(ii) A vector joining two points of the reciprocal lattice, G, is perpendicular to the
corresponding plane of the real lattice (Figure 2.38). For instance, a vector
joining points (1, 1, 1) and (0, 0, 0) in reciprocal space will be perpendicular to
the {111} planes in real space, with a magnitude of 1/d 111 . As you might expect,
one may easily determine the interplanar spacing for sets of real lattice planes
by taking the inverse of that represented in Eq. 20 – e.g., the distance between
adjacent planes of {100} in the real lattice is:

½a ðb cފ
ð21Þ
j b cj
In order to more completely understand reciprocal space relative to “real” space
defined by Cartesian coordinates, we need to first recall the deBroglie relationship
that describes the wave-particle duality of matter:
h h
ð22Þ l ¼ ,or p¼ ;
l
p